Bimodules and natural transformations for enriched $\infty$-categories

TL;DR

This paper develops a framework for bimodules and natural transformations within enriched $$-categories, constructing a double $$-category and clarifying the role of natural transformations in this setting.

Contribution

It introduces a notion of bimodule for enriched $$-categories and constructs a double $$-category incorporating functors and bimodules, providing a natural definition of transformations.

Findings

Bimodules are formalized for enriched $$-categories.

A double $$-category of enriched $$-categories is constructed.

Natural transformations are characterized as 2-morphisms in the $(,2)$-category.

Abstract

We introduce a notion of bimodule in the setting of enriched $\infty$-categories, and use this to construct a double $\infty$-category of enriched $\infty$-categories where the two kinds of 1-morphisms are functors and bimodules. We then consider a natural definition of natural transformations in this context, and show that in the underlying $(\infty,2)$-category of enriched $\infty$-categories with functors as 1-morphisms the 2-morphisms are given by natural transformations.